61 Large Sample Confidence Intervals for a Mean A confidence interval for a parameter is a databased interval of numbers likely to include the true value of the parameter with a probabilitybased confidence ID: 578984
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Slide1
Ch 6 Introduction to Formal Statistical InferenceSlide2
6.1 Large Sample Confidence Intervals for a Mean
A confidence interval for a parameter is a data-based interval of numbers likely to include the true value of the parameter with a probability-based confidence.
A 95% confidence interval for µ is an interval which was constructed in a manner such that 95% of such intervals contain the true value of µ.Slide3
Interval Estimate—Confidence intervals
An interval estimate consists of an interval which will contain the quantity it is supposed to estimate with a specified probability (or degree of confidence).
Recall that for large random
samples, the
sampling distribution of the mean is approximately a normal distribution with So we will utilize some properties of normal distribution to explain a confidence interval.Slide4
For a standard normal curveSlide5
Large-sample known
s
confidence interval for
m.Slide6
Confidence Intervals
100(1-a)% CI:
80%
90%
95%
99% Slide7
Confidence Interval for Means
After computing sample mean , find a range of values such that 95% of the time the resulting range includes the true value
m
.Slide8Slide9
X=breaking strength of a fish line.
σ
=0.10. In a random sample of size n=10,
Find a 95% confidence interval for μ, the true average breaking strength. Slide10Slide11Slide12Slide13
How large a sample size is needed in order to get
an error of no
more than
0.01 with
95% probability if the sample mean is used to estimate the true mean?Solution n=385, always round up!Slide14
Example
A certain adjustment to a machine will change the length of the parts it is making but will not affect the standard deviation.
The length of the parts is normally distributed, and the standard deviation is 0.5 mm (millimeter).
After an adjustment is made, a random sample is taken to determine the mean length of parts now being produced. The observed lengths are
75.3, 76.0, 75.0, 77.0, 75.4, 76.3, 77.0, 74.9, 76.5, 75.8. Slide15
Questions
What is the parameter of interest?
Find the point estimate of the mean length of parts now being produced.
c. Find the 99% confidence interval for μ.
d. How large a sample should be taken if the population mean is to be estimated with 99% confidence to have an error not exceeding 0.2 mm ?Slide16
Solution
a.
The mean length of parts now being produced (μ);
b. x=75.92c. n=10; σ=0.5; . The 99% confidence interval is
75.512<μ<76.328
Δ
=0.20
;
since n must be an integer, n=42.